A Comparison of Hierarchical Bayesian Models for Small Area Estimation of Counts

Small area estimation (SAE) tackles the problem of providing reliable estimates for small areas, i.e., subsets of the population for which sample information is not sufficient to warrant the use of a direct estimator. Hierarchical Bayesian approach to SAE problems offers several advantages over traditional SAE models including the ability of appropriately accounting for the type of surveyed variable. In this paper, a number of model specifications for estimating small area counts are discussed and their relative merits are illustrated. We conducted a simulation study by reproducing in a simplified form the Italian Labour Force Survey and taking the Local Labor Markets as target areas. Simulated data were generated by assuming population characteristics of interest as well as survey sampling design as known. In one set of experiments, numbers of employment/unemployment from census data were utilized, in others population characteristics were varied. Results show persistent model failures for some standard Fay-Herriot specifications and for generalized linear Poisson models with (log-)normal sampling stage, whilst either unmatched or nonnormal sampling stage models get the best performance in terms of bias, accuracy and reliability. Though, the study also found that any model noticeably improves on its performance by letting sampling variances be stochastically determined rather than assumed as known as is the general practice. Moreover, we address the issue of model determination to point out limits and possible deceptions of commonly used criteria for model selection and checking in SAE context.

Local and regional flood frequency analysis based on hierarchical Bayesian model in Dongting Lake Basin,China

This study developed a hierarchical Bayesian(HB)model for local and regional flood frequency analysis in the Dongting Lake Basin,in China.The annual maximum daily flows from 15 streamflow-gauged sites in the study area were analyzed with the HB model.The generalized extreme value(GEV)distribution was selected as the extreme flood distribution,and the GEV distribution location and scale parameters were spatially modeled through a regression approach with the drainage area as a covariate.The Markov chain Monte Carlo(MCMC)method with Gibbs sampling was employed to calculate the posterior distribution in the HB model.The results showed that the proposed HB model provided satisfactory Bayesian credible intervals for flood quantiles,while the traditional delta method could not provide reliable uncertainty estimations for large flood quantiles,due to the fact that the lower confidence bounds tended to decrease as the return periods increased.Furthermore,the HB model for regional analysis allowed for a reduction in the value of some restrictive assumptions in the traditional index flood method,such as the homogeneity region assumption and the scale invariance assumption.The HB model can also provide an uncertainty band of flood quantile prediction at a poorly gauged or ungauged site,but the index flood method with L-moments does not demonstrate this uncertainty directly.Therefore,the HB model is an effective method of implementing the flexible local and regional frequency analysis scheme,and of quantifying the associated predictive uncertainty.

A Bayesian hierarchical model for the inference between metal grade with reduced variance:Case studies in porphyry Cu deposits

Ore sorting is a preconcentration technology and can dramatically reduce energy and water usage to improve the sustainability and profitability of a mining operation.In porphyry Cu deposits,Cu is the primary target,with ores usually containing secondary‘pay’metals such as Au,Mo and gangue elements such as Fe and As.Due to sensing technology limitations,secondary and deleterious materials vary in correlation type and strength with Cu but cannot be detected simultaneously via magnetic resonance(MR)ore sorting.Inferring the relationships between Cu and other elemental abundances is particularly critical for mineral processing.The variations in metal grade relationships occur due to the transition into different geological domains.This raises two questions-how to define these geological domains and how the metal grade relationship is influenced by these geological domains.In this paper,linear relationship is assumed between Cu grade and other metal grades.We applies a Bayesian hierarchical(partial-pooling)model to quantify the linear relationships between Cu,Au,and Fe grades from geochemical bore core data.The hierarchical model was compared with two other models-‘complete-pooling’model and‘nopooling’model.Mining blocks were split based on spatial domain to construct hierarchical model.Geochemical bore core data records metal grades measured from laboratory assay with spatial coordinates of sample location.Two case studies from different porphyry Cu deposits were used to evaluate the performance of the hierarchical model.Markov chain Monte Carlo(MCMC)was used to sample the posterior parameters.Our results show that the Bayesian hierarchical model dramatically reduced the posterior predictive variance for metal grades regression compared to the no-pooling model.In addition,the posterior inference in the hierarchical model is insensitive to the choice of prior.The data is wellrepresented in the posterior which indicates a robust model.The results show that the spatial domain can be successfully utilised for metal grade regression.Uncertainty in estimating the relationship between pay metals and both secondary and gangue elements is quantified and shown to be reduced with partial-pooling.Thus,the proposed Bayesian hierarchical model can offer a reliable and stable way to monitor the relationship between metal grades for ore sorting and other mineral processing options.

Bayesian partial pooling to reduce uncertainty in overcoring rock stress estimation

The state of in situ stress is a crucial parameter in subsurface engineering,especially for critical projects like nuclear waste repository.As one of the two ISRM suggested methods,the overcoring(OC)method is widely used to estimate the full stress tensors in rocks by independent regression analysis of the data from each OC test.However,such customary independent analysis of individual OC tests,known as no pooling,is liable to yield unreliable test-specific stress estimates due to various uncertainty sources involved in the OC method.To address this problem,a practical and no-cost solution is considered by incorporating into OC data analysis additional information implied within adjacent OC tests,which are usually available in OC measurement campaigns.Hence,this paper presents a Bayesian partial pooling(hierarchical)model for combined analysis of adjacent OC tests.We performed five case studies using OC test data made at a nuclear waste repository research site of Sweden.The results demonstrate that partial pooling of adjacent OC tests indeed allows borrowing of information across adjacent tests,and yields improved stress tensor estimates with reduced uncertainties simultaneously for all individual tests than they are independently analysed as no pooling,particularly for those unreliable no pooling stress estimates.A further model comparison shows that the partial pooling model also gives better predictive performance,and thus confirms that the information borrowed across adjacent OC tests is relevant and effective.

Construct Validation by Hierarchical Bayesian Concept Maps: An Application to the Transaction Cost Economics Theory of the Firm

A concept map is a diagram depicting relationships among concepts which is used as a knowledge representation tool in many knowledge domains. In this paper, we build on the modeling framework of Hui et al. (2008) in order to develop a concept map suitable for testing the empirical evidence of theories. We identify a theory by a set of core tenets each asserting that one set of independent variables affects one dependent variable, moreover every variable can have several operational definitions. Data consist of a selected sample of scientific articles from the empirical literature on the theory under investigation. Our “tenet map” features a number of complexities more than the original version. First the links are two-layer: first-layer links connect variables which are related in the test of the theory at issue;second-layer links represent connections which are found statistically significant. Besides, either layer matrix of link-formation probabilities is block-symmetric. In addition to a form of censoring which resembles the Hui et al. pruning step, observed maps are subject to a further censoring related to second-layer links. Still, we perform a full Bayesian analysis instead of adopting the empirical Bayes approach. Lastly, we develop a three-stage model which accounts for dependence either of data or of parameters. The investigation of the empirical support and consensus degree of new economic theories of the firm motivated the proposed methodology. In this paper, the Transaction Cost Economics view is tested by a tenet map analysis. Both the two-stage and the multilevel models identify the same tenets as the most corroborated by empirical evidence though the latter provides a more comprehensive and complex insight of relationships between constructs.

Bayesian Segmentation of Piecewise Linear Regression Models Using Reversible Jump MCMC Algorithm

Piecewise linear regression models are very flexible models for modeling the data. If the piecewise linear regression models are matched against the data, then the parameters are generally not known. This paper studies the problem of parameter estimation ofpiecewise linear regression models. The method used to estimate the parameters ofpicewise linear regression models is Bayesian method. But the Bayes estimator can not be found analytically. To overcome these problems, the reversible jump MCMC （Marcov Chain Monte Carlo） algorithm is proposed. Reversible jump MCMC algorithm generates the Markov chain converges to the limit distribution of the posterior distribution of the parameters ofpicewise linear regression models. The resulting Markov chain is used to calculate the Bayes estimator for the parameters of picewise linear regression models.

An application of Bayesian multilevel model to evaluate variations in stochastic and dynamic transition of traffic conditions

This study seeks to investigate the variations associated with lane lateral locations and days of the week in the stochastic and dynamic transition of traffic regimes(DTTR).In the proposed analysis,hierarchical regression models fitted using Bayesian frameworks were used to calibrate the transition probabilities that describe the DTTR.Datasets of two sites on a freeway facility located in Jacksonville,Florida,were selected for the analysis.The traffic speed thresholds to define traffic regimes were estimated using the Gaussian mixture model(GMM).The GMM revealed that two and three regimes were adequate mixture components for estimating the traffic speed distributions for Site 1 and 2 datasets,respectively.The results of hierarchical regression models show that there is considerable evidence that there are heterogeneity characteristics in the DTTR associated with lateral lane locations.In particular,the hierarchical regressions reveal that the breakdown process is more affected by the variations compared to other evaluated transition processes with the estimated intra-class correlation(ICC)of about 73%.The transition from congestion on-set/dissolution(COD)to the congested regime is estimated with the highest ICC of 49.4%in the three-regime model,and the lowest ICC of 1%was observed on the transition from the congested to COD regime.On the other hand,different days of the week are not found to contribute to the variations(the highest ICC was 1.44%)on the DTTR.These findings can be used in developing effective congestion countermeasures,particularly in the application of intelligent transportation systems,such as dynamic lane-management strategies.

Minimum Description Length Methods in Bayesian Model Selection: Some Applications

Computations involved in Bayesian approach to practical model selection problems are usually very difficult. Computational simplifications are sometimes possible, but are not generally applicable. There is a large literature available on a methodology based on information theory called Minimum Description Length (MDL). It is described here how many of these techniques are either directly Bayesian in nature, or are very good objective approximations to Bayesian solutions. First, connections between the Bayesian approach and MDL are theoretically explored;thereafter a few illustrations are provided to describe how MDL can give useful computational simplifications.

基于贝叶斯分层模型的液化侧移稳健的易损性分析方法

提出了一种基于贝叶斯分层模型的液化侧移稳健的易损性分析方法。采用贝叶斯分层模型量化不同增量动力分析(IDA)曲线间的差异,结合抽样方法预测潜在侧移的分布,建立液化侧移稳健的易损性曲线和超越概率曲线。以一处实际发生过液化侧移的场地为例,展示了稳健的易损性曲线及超越概率曲线的建立方法,并与相关方法进行比较。结果表明,所提出的方法可以较好地模拟IDA曲线的分布,较为准确地量化易损性曲线和超越概率曲线的不确定性。

基于BHM-EcoFlow模型的汉江中下游河段水文-生态响应关系研究

河流水文-生态响应关系是确定生态流量阈值的科学基础。针对当前河流水文-生态响应关系研究中生态数据不足且生态建模难度大的问题,建立了基于贝叶斯层次分析法的BHM-EcoFlow(Bayesian hierarchical modelling-ecological flow)模型,该模型将河流不同河段及同一河段不同站点间的先验知识与实测数据相结合,可有效利用短系列数据,实现河流水文-生态响应关系的模拟。采用汉江中下游干流2011年的水文、生态数据,模拟了浮游植物细胞密度与流量、混合层温度间的关系,计算了不同流量条件下各河段的浮游植物密度。结果表明:BHM-EcoFlow模型提高了短系列数据的可用性,对汉江中下游干流的水文-生态响应关系具有良好的识别能力,为确定生态流量提供了科学依据。

Bayesian两变量层次模型及其在诊断试验系统评价中的应用

目的探讨Bayesian两变量层次模型的构建及其在诊断试验系统评价中的应用。方法将Bayesian两变量层次模型应用于传统Pap细胞学涂片诊断子宫颈癌准确性评价的历史Meta分析资料,估计相关的效应指标敏感度和特异度及筛查研究比随访研究的相对可信度。结果与经典综合受试者工作特征曲线方法相比,Bayesian两变量层次模型估计得到三个层次的效应指标,其中综合敏感度和特异度均数及95%可信区间分别为0.64(0.56,0.72)和0.74(0.67,0.80),预测敏感度和特异度均数及95%可信区间分别为0.61(0.12,0.96)和0.69(0.21,0.97),筛查研究比随访研究的相对可信度估计为1.3(0.59,2.48)。结论采用Bayesian两变量层次模型进行诊断试验Meta分析,更加灵活、有效,易于实现和解释,值得推广应用。

基于贝叶斯层级模型的用户异常行为检测研究

大多数操作系统的安全防护主要依赖基于签名或基于规则的方法,因此现有大多数的异常检测方法精度较低。因此,利用贝叶斯模型为同类群体建模,并结合时间效应与分层原则,为用户实体行为分析(User and Entity Behavior Analytics,UEBA)研究提供精度更高的数据集。然后,将基于实际记录的用户行为数据与贝叶斯层级图模型推测出的数据进行比较,降低模型中的误报率。该方法主要分为两个阶段:在第1阶段,基于数据驱动的方法形成用户行为聚类,定义用户的个人身份验证模式;在第2阶段,同时考虑到周期性因素和分层原则,并通过泊松分布建模。研究表明,数据驱动的聚类方法在减少误报方面能够取得更好的结果,并减轻网络安全管理的负担,进一步减少误报数量。

基于有偏误辅助变量的分层贝叶斯小域估计方法研究

抽样调查中的小域估计问题指的是,根据较少样本量进行一定精度下子总体估计的现实问题。与基于设计的方法不同,基于模型的方法不依赖大样本理论,能在估计过程中借助其他域的样本信息,更加适用于小域估计问题。然而,现实中测量误差无法完全避免,当模型协变量有偏误时,小域估计结果失效。对此,采用测量误差模型校正辅助变量误差,基于单元层次的分层贝叶斯模型进行小域估计,并在贝叶斯框架下估计辅助变量偏误机制。鉴于实际调查中为方便数据编码与统计、控制无回答误差,调查结果以分类型数据居多,本文重点讨论了更适用于小域估计问题的模型方法,针对分类型辅助变量存在测量误差的情形,给出了方法合理性的证明,同时通过模拟和实证对其估计效果进行验证与实践。本文模拟六种实践中常见的情形,除仅有分类型变量存在测量误差的情形之外,还考虑了存在测量误差的变量既有分类型又有连续型的情形等。数值模拟与实证结果一致表明,本文方法不仅能充分纳入与推断相关的不确定性因素,克服样本量受限的问题,还具有广泛的适用性,相较于传统方法,估计结果在提升准确度的同时更为稳健。

基于贝叶斯分层模型的低复杂度无线传感器网络定位算法

文章对基于压缩感知的无线传感器网络定位算法进行了研究,存在重构算法计算量大、定位误差较大等问题,为降低计算复杂度和定位误差,文章提出基于贝叶斯分层模型的低复杂度无线传感器网络定位算法。首先,将稀疏贝叶斯分层先验模型引入到无线传感器网络的定位中;其次,通过运用稀疏贝叶斯理论推理出估计目标的后验概率分布;最后,结合变分消息传递(VMP)算法,使用辅助函数对未知变量的联合后验概率密度函数进行等效,得到目标位置向量的估计结果。仿真结果表明,相较于传统的重构算法,文章提出的方法具有更好的恢复效果,计算复杂度更低。

基于HHM-RFRM的海上风电工程吊装作业风险评估研究

为准确、全面识别海上风电工程吊装作业风险,分析施工过程中的风险交互作用,基于等级全息建模思想与风险过滤、评级与管理(HHM-RFRM)理论,建立海上风电吊装施工多维风险情景度量化方法。首先,从人、环、机、管4个方面构建全息评估指标体系;其次,采用RFRM法对风险因素进行量化排序;最后,依据贝叶斯公式构建二维典型风险场景。结果表明:环境因素对吊装作业安全影响最大,潮湿作业环境分别与作业船稳定性差、吊索具出现故障以及操作人员配合不当等3个因素交互时事故风险度高。该方法可为海上风电施工安全风险的精准管控提供理论依据。

Population dynamics modelling with spatial heterogeneity for yellow croaker(Larimichthys polyactis)along the coast of China

As one of the top four commercially important species in China,yellow croaker(Larimichthys polyactis)with two geographic subpopulations,has undergone profound changes during the last several decades.It is widely comprehended that understanding its population dynamics is critically important for sustainable management of this valuable fishery in China.The only two existing population dynamics models assessed the population of yellow croaker using short time-series data,without considering geographical variations.In this study,Bayesian models with and without hierarchical subpopulation structure were developed to explore the spatial heterogeneity of the population dynamics of yellow croaker from 1968 to 2015.Alternative hypotheses were constructed to test potential temporal patterns in yellow croaker’s population dynamics.Substantial variations in population dynamics characteristics among space and time were found through this study.The population growth rate was revealed to increase since the late 1980s,and the catchability increased more than twice from 1981 to 2015.The East China Sea’s subpopulation witnesses faster growth,but suffers from higher fishing pressure than that in the Bohai Sea and Yellow Sea.The global population and two subpopulations all have high risks of overfishing and being overfished according to the MSY-based reference points in recent years.More conservative management strategies with subpopulation considerations are imperative for the fishery management of yellow croaker in China.The methodology developed in this study could also be applied to the stock assessment and fishery management of other species,especially for those species with large spatial heterogeneity data.

Bayesian meta-analysis of regional biomass factors for Quercus mongolica forests in South Korea

Indirect approaches to estimation of biomass factors are often applied to measure carbon flux in the forestry sector. An assumption underlying a country-level carbon stock estimate is the representativeness of these factors. Although intensive studies have been conducted to quantify biomass factors, each study typically covers a limited geographic area. The goal of this study was to employ a meta-analysis approach to develop regional bio- mass factors for Quercus mongolica forests in South Korea. The biomass factors of interest were biomass conversion and expansion factor （BCEF）, biomass expansion factor （BEF） and root-to-shoot ratio （RSR）. Our objectives were to select probability density functions （PDFs） that best fitted the three biomass factors and to quantify their means and uncertainties. A total of 12 scientific publications were selected as data sources based on a set of criteria. Fromthese publications we chose 52 study sites spread out across South Korea. The statistical model for the meta- analysis was a multilevel model with publication （data source） as the nesting factor specified under the Bayesian framework. Gamma, Log-normal and Weibull PDFs were evaluated. The Log-normal PDF yielded the best quanti- tative and qualitative fit for the three biomass factors. However, a poor fit of the PDF to the long right tail of observed BEF and RSR distributions was apparent. The median posterior estimates for means and 95 % credible intervals for BCEF, BEF and RSR across all 12 publica- tions were 1.016 （0.800-1.299）, 1.414 （1.304-1.560） and 0.260 （0.200-0.335）, respectively. The Log-normal PDF proved useful for estimating carbon stock of Q. mongolica forests on a regional scale and for uncertainty analysis based on Monte Carlo simulation.

Posterior propriety of an objective prior for generalized hierarchical normal linear models

Bayesian Hierarchical models has been widely used in modern statistical application.To deal with the data having complex structures,we propose a generalized hierarchical normal linear(GHNL)model which accommodates arbitrarily many levels,usual design matrices and'vanilla'covari-ance matrices.Objective hyperpriors can be employed for the GHNL model to express ignorance or match frequentist properties,yet the common objective Bayesian approaches are infeasible or fraught with danger in hierarchical modelling.To tackle this issue,[Berger,J,Sun,D.&Song,C.(2020b).An objective prior for hyperparameters in normal hierarchical models.Journal of Multi-variate Analysis,178,104606.https://doi.org/10.1016/jmva.2020.104606]proposed a particular objective prior and investigated its properties comprehensively.Posterior propriety is important for the choice of priors to guarantee the convergence of MCMC samplers.James Berger conjec-tured that the resulting posterior is proper for a hierarchical normal model with arbitrarily many levels,a rigorous proof of which was not given,however.In this paper,we complete this story and provide an user friendly guidance.One main contribution of this paper is to propose a new technique for deriving an elaborate upper bound on the integrated likelihood but also one uni-fied approach to checking the posterior propriety for linear models.An eficient Gibbs sampling method is also introduced and outperforms other sampling approaches considerably.

Integration of expression profiles and endo-phenotypes in genetic association studies: A Bayesian approach to determine the path from gene to disease

In genetic association studies of complex diseases, endo-phenotypes such as expression profiles, epigenetic data, or clinical intermediate-phenotypes provide insight to understand the underlying biological path of the disease. In such situations, in order to establish the path from the gene to the disease, we have to decide whether the gene acts on the disease phenotype primarily through a specific endo-phenotype or whether the gene influences the disease through an unidentified path which is characterized by different intermediate phenotypes. Here, we address the question that a genetic locus, given its effect on an endo-phenotype, influences the trait of interest primarily through the path of the endo-phenotype. We propose a Bayesian approach that can evaluate the genetic association between the genetic locus and the phenotype of interest in the presence of the genetic effect on the endo-phenotype. Using simulation studies, we verify that our approach has the desired properties and compare this approach with a mediation approach. The proposed Bayesian approach is illustrated by an application to genome-wide association study for childhood asthma (CAMP) that contains expression profiles.

Portfolio optimisation using constrained hierarchical bayes models

It iswell known that traditionalmean-variance optimal portfolio delivers rather erratic and unsatisfactory out-of-sample performance due to the neglect of estimation errors.Constrained solutions,such as no-short-sale-constrained and norm-constrained portfolios,can usually achieve much higher ex post Sharpe ratio.Bayesian methods have also been shown to be superior to traditional plug-in estimator by incorporating parameter uncertainty through prior distributions.In this paper,we develop an innovative method that induces priors directly on optimal portfolio weights and imposing constraints a priori in our hierarchical Bayes model.We showthat such constructed portfolios are well diversified with superior out-of-sample performance.Our proposed model is tested on a number of Fama–French industry portfolios against the na飗e diversification strategy and Chevrier and McCulloch’s(2008)economically motivated prior(EMP)strategy.On average,our model outperforms Chevrier and McCulloch’s(2008)EMP strategy by over 15%and outperform the‘1/N’strategy by over 50%.